LMFDB - Embedded newform 6498.2.a.bk.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6498,2,Mod(1,6498)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6498, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6498.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6498 = 2 \cdot 3^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6498.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$51.8867912334$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 722)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	6498.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.38197 q^{5} +1.23607 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} +1.38197 q^{5} +1.23607 q^{7} +1.00000 q^{8} +1.38197 q^{10} -1.23607 q^{11} +3.61803 q^{13} +1.23607 q^{14} +1.00000 q^{16} +5.61803 q^{17} +1.38197 q^{20} -1.23607 q^{22} -0.763932 q^{23} -3.09017 q^{25} +3.61803 q^{26} +1.23607 q^{28} -2.09017 q^{29} +3.23607 q^{31} +1.00000 q^{32} +5.61803 q^{34} +1.70820 q^{35} +10.6180 q^{37} +1.38197 q^{40} +5.85410 q^{41} -4.76393 q^{43} -1.23607 q^{44} -0.763932 q^{46} +4.47214 q^{47} -5.47214 q^{49} -3.09017 q^{50} +3.61803 q^{52} -5.09017 q^{53} -1.70820 q^{55} +1.23607 q^{56} -2.09017 q^{58} -8.47214 q^{59} +3.61803 q^{61} +3.23607 q^{62} +1.00000 q^{64} +5.00000 q^{65} -1.70820 q^{67} +5.61803 q^{68} +1.70820 q^{70} +14.9443 q^{71} -3.38197 q^{73} +10.6180 q^{74} -1.52786 q^{77} -7.23607 q^{79} +1.38197 q^{80} +5.85410 q^{82} -8.47214 q^{83} +7.76393 q^{85} -4.76393 q^{86} -1.23607 q^{88} -2.14590 q^{89} +4.47214 q^{91} -0.763932 q^{92} +4.47214 q^{94} +6.38197 q^{97} -5.47214 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 5 q^{5} - 2 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 + 5 * q^5 - 2 * q^7 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} + 5 q^{5} - 2 q^{7} + 2 q^{8} + 5 q^{10} + 2 q^{11} + 5 q^{13} - 2 q^{14} + 2 q^{16} + 9 q^{17} + 5 q^{20} + 2 q^{22} - 6 q^{23} + 5 q^{25} + 5 q^{26} - 2 q^{28} + 7 q^{29} + 2 q^{31} + 2 q^{32} + 9 q^{34} - 10 q^{35} + 19 q^{37} + 5 q^{40} + 5 q^{41} - 14 q^{43} + 2 q^{44} - 6 q^{46} - 2 q^{49} + 5 q^{50} + 5 q^{52} + q^{53} + 10 q^{55} - 2 q^{56} + 7 q^{58} - 8 q^{59} + 5 q^{61} + 2 q^{62} + 2 q^{64} + 10 q^{65} + 10 q^{67} + 9 q^{68} - 10 q^{70} + 12 q^{71} - 9 q^{73} + 19 q^{74} - 12 q^{77} - 10 q^{79} + 5 q^{80} + 5 q^{82} - 8 q^{83} + 20 q^{85} - 14 q^{86} + 2 q^{88} - 11 q^{89} - 6 q^{92} + 15 q^{97} - 2 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 5 * q^5 - 2 * q^7 + 2 * q^8 + 5 * q^10 + 2 * q^11 + 5 * q^13 - 2 * q^14 + 2 * q^16 + 9 * q^17 + 5 * q^20 + 2 * q^22 - 6 * q^23 + 5 * q^25 + 5 * q^26 - 2 * q^28 + 7 * q^29 + 2 * q^31 + 2 * q^32 + 9 * q^34 - 10 * q^35 + 19 * q^37 + 5 * q^40 + 5 * q^41 - 14 * q^43 + 2 * q^44 - 6 * q^46 - 2 * q^49 + 5 * q^50 + 5 * q^52 + q^53 + 10 * q^55 - 2 * q^56 + 7 * q^58 - 8 * q^59 + 5 * q^61 + 2 * q^62 + 2 * q^64 + 10 * q^65 + 10 * q^67 + 9 * q^68 - 10 * q^70 + 12 * q^71 - 9 * q^73 + 19 * q^74 - 12 * q^77 - 10 * q^79 + 5 * q^80 + 5 * q^82 - 8 * q^83 + 20 * q^85 - 14 * q^86 + 2 * q^88 - 11 * q^89 - 6 * q^92 + 15 * q^97 - 2 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.38197	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$5$	1.38197	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$6$	0	0
$7$	1.23607	0.467190	0.233595	−	0.972334i	$-0.424951\pi$
$7$	1.23607	0.467190	0.233595	+	0.972334i	$0.424951\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	1.38197	0.437016
$11$	−1.23607	−0.372689	−0.186344	−	0.982485i	$-0.559664\pi$
$11$	−1.23607	−0.372689	−0.186344	+	0.982485i	$0.559664\pi$
$12$	0	0
$13$	3.61803	1.00346	0.501731	−	0.865024i	$-0.332697\pi$
$13$	3.61803	1.00346	0.501731	+	0.865024i	$0.332697\pi$
$14$	1.23607	0.330353
$15$	0	0
$16$	1.00000	0.250000
$17$	5.61803	1.36257	0.681287	−	0.732017i	$-0.261421\pi$
$17$	5.61803	1.36257	0.681287	+	0.732017i	$0.261421\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	1.38197	0.309017
$21$	0	0
$22$	−1.23607	−0.263531
$23$	−0.763932	−0.159291	−0.0796454	−	0.996823i	$-0.525379\pi$
$23$	−0.763932	−0.159291	−0.0796454	+	0.996823i	$0.525379\pi$
$24$	0	0
$25$	−3.09017	−0.618034
$26$	3.61803	0.709555
$27$	0	0
$28$	1.23607	0.233595
$29$	−2.09017	−0.388135	−0.194067	−	0.980988i	$-0.562168\pi$
$29$	−2.09017	−0.388135	−0.194067	+	0.980988i	$0.562168\pi$
$30$	0	0
$31$	3.23607	0.581215	0.290607	−	0.956842i	$-0.406143\pi$
$31$	3.23607	0.581215	0.290607	+	0.956842i	$0.406143\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	5.61803	0.963485
$35$	1.70820	0.288739
$36$	0	0
$37$	10.6180	1.74559	0.872797	−	0.488083i	$-0.162304\pi$
$37$	10.6180	1.74559	0.872797	+	0.488083i	$0.162304\pi$
$38$	0	0
$39$	0	0
$40$	1.38197	0.218508
$41$	5.85410	0.914257	0.457129	−	0.889401i	$-0.348878\pi$
$41$	5.85410	0.914257	0.457129	+	0.889401i	$0.348878\pi$
$42$	0	0
$43$	−4.76393	−0.726493	−0.363246	−	0.931693i	$-0.618332\pi$
$43$	−4.76393	−0.726493	−0.363246	+	0.931693i	$0.618332\pi$
$44$	−1.23607	−0.186344
$45$	0	0
$46$	−0.763932	−0.112636
$47$	4.47214	0.652328	0.326164	−	0.945313i	$-0.394244\pi$
$47$	4.47214	0.652328	0.326164	+	0.945313i	$0.394244\pi$
$48$	0	0
$49$	−5.47214	−0.781734
$50$	−3.09017	−0.437016
$51$	0	0
$52$	3.61803	0.501731
$53$	−5.09017	−0.699189	−0.349594	−	0.936901i	$-0.613681\pi$
$53$	−5.09017	−0.699189	−0.349594	+	0.936901i	$0.613681\pi$
$54$	0	0
$55$	−1.70820	−0.230334
$56$	1.23607	0.165177
$57$	0	0
$58$	−2.09017	−0.274453
$59$	−8.47214	−1.10298	−0.551489	−	0.834182i	$-0.685940\pi$
$59$	−8.47214	−1.10298	−0.551489	+	0.834182i	$0.685940\pi$
$60$	0	0
$61$	3.61803	0.463242	0.231621	−	0.972806i	$-0.425597\pi$
$61$	3.61803	0.463242	0.231621	+	0.972806i	$0.425597\pi$
$62$	3.23607	0.410981
$63$	0	0
$64$	1.00000	0.125000
$65$	5.00000	0.620174
$66$	0	0
$67$	−1.70820	−0.208690	−0.104345	−	0.994541i	$-0.533275\pi$
$67$	−1.70820	−0.208690	−0.104345	+	0.994541i	$0.533275\pi$
$68$	5.61803	0.681287
$69$	0	0
$70$	1.70820	0.204169
$71$	14.9443	1.77356	0.886779	−	0.462193i	$-0.152937\pi$
$71$	14.9443	1.77356	0.886779	+	0.462193i	$0.152937\pi$
$72$	0	0
$73$	−3.38197	−0.395829	−0.197915	−	0.980219i	$-0.563417\pi$
$73$	−3.38197	−0.395829	−0.197915	+	0.980219i	$0.563417\pi$
$74$	10.6180	1.23432
$75$	0	0
$76$	0	0
$77$	−1.52786	−0.174116
$78$	0	0
$79$	−7.23607	−0.814121	−0.407061	−	0.913401i	$-0.633446\pi$
$79$	−7.23607	−0.814121	−0.407061	+	0.913401i	$0.633446\pi$
$80$	1.38197	0.154508
$81$	0	0
$82$	5.85410	0.646477
$83$	−8.47214	−0.929938	−0.464969	−	0.885327i	$-0.653934\pi$
$83$	−8.47214	−0.929938	−0.464969	+	0.885327i	$0.653934\pi$
$84$	0	0
$85$	7.76393	0.842117
$86$	−4.76393	−0.513708
$87$	0	0
$88$	−1.23607	−0.131765
$89$	−2.14590	−0.227465	−0.113732	−	0.993511i	$-0.536281\pi$
$89$	−2.14590	−0.227465	−0.113732	+	0.993511i	$0.536281\pi$
$90$	0	0
$91$	4.47214	0.468807
$92$	−0.763932	−0.0796454
$93$	0	0
$94$	4.47214	0.461266
$95$	0	0
$96$	0	0
$97$	6.38197	0.647990	0.323995	−	0.946059i	$-0.394974\pi$
$97$	6.38197	0.647990	0.323995	+	0.946059i	$0.394974\pi$
$98$	−5.47214	−0.552769
$99$	0	0
$100$	−3.09017	−0.309017
$101$	13.3820	1.33156	0.665778	−	0.746150i	$-0.268100\pi$
$101$	13.3820	1.33156	0.665778	+	0.746150i	$0.268100\pi$
$102$	0	0
$103$	−16.6525	−1.64082	−0.820409	−	0.571778i	$-0.806254\pi$
$103$	−16.6525	−1.64082	−0.820409	+	0.571778i	$0.806254\pi$
$104$	3.61803	0.354777
$105$	0	0
$106$	−5.09017	−0.494401
$107$	17.4164	1.68371	0.841854	−	0.539706i	$-0.181464\pi$
$107$	17.4164	1.68371	0.841854	+	0.539706i	$0.181464\pi$
$108$	0	0
$109$	18.0344	1.72739	0.863693	−	0.504018i	$-0.168145\pi$
$109$	18.0344	1.72739	0.863693	+	0.504018i	$0.168145\pi$
$110$	−1.70820	−0.162871
$111$	0	0
$112$	1.23607	0.116797
$113$	3.14590	0.295941	0.147971	−	0.988992i	$-0.452726\pi$
$113$	3.14590	0.295941	0.147971	+	0.988992i	$0.452726\pi$
$114$	0	0
$115$	−1.05573	−0.0984472
$116$	−2.09017	−0.194067
$117$	0	0
$118$	−8.47214	−0.779923
$119$	6.94427	0.636580
$120$	0	0
$121$	−9.47214	−0.861103
$122$	3.61803	0.327561
$123$	0	0
$124$	3.23607	0.290607
$125$	−11.1803	−1.00000
$126$	0	0
$127$	15.4164	1.36798	0.683992	−	0.729489i	$-0.260242\pi$
$127$	15.4164	1.36798	0.683992	+	0.729489i	$0.260242\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	5.00000	0.438529
$131$	4.47214	0.390732	0.195366	−	0.980730i	$-0.437410\pi$
$131$	4.47214	0.390732	0.195366	+	0.980730i	$0.437410\pi$
$132$	0	0
$133$	0	0
$134$	−1.70820	−0.147566
$135$	0	0
$136$	5.61803	0.481742
$137$	−4.61803	−0.394545	−0.197273	−	0.980349i	$-0.563208\pi$
$137$	−4.61803	−0.394545	−0.197273	+	0.980349i	$0.563208\pi$
$138$	0	0
$139$	−14.0000	−1.18746	−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000	−1.18746	−0.593732	+	0.804663i	$0.702346\pi$
$140$	1.70820	0.144370
$141$	0	0
$142$	14.9443	1.25410
$143$	−4.47214	−0.373979
$144$	0	0
$145$	−2.88854	−0.239881
$146$	−3.38197	−0.279893
$147$	0	0
$148$	10.6180	0.872797
$149$	1.90983	0.156459	0.0782297	−	0.996935i	$-0.475073\pi$
$149$	1.90983	0.156459	0.0782297	+	0.996935i	$0.475073\pi$
$150$	0	0
$151$	5.52786	0.449851	0.224926	−	0.974376i	$-0.427786\pi$
$151$	5.52786	0.449851	0.224926	+	0.974376i	$0.427786\pi$
$152$	0	0
$153$	0	0
$154$	−1.52786	−0.123119
$155$	4.47214	0.359211
$156$	0	0
$157$	11.6180	0.927220	0.463610	−	0.886039i	$-0.346554\pi$
$157$	11.6180	0.927220	0.463610	+	0.886039i	$0.346554\pi$
$158$	−7.23607	−0.575671
$159$	0	0
$160$	1.38197	0.109254
$161$	−0.944272	−0.0744191
$162$	0	0
$163$	−20.6525	−1.61763	−0.808813	−	0.588065i	$-0.799890\pi$
$163$	−20.6525	−1.61763	−0.808813	+	0.588065i	$0.799890\pi$
$164$	5.85410	0.457129
$165$	0	0
$166$	−8.47214	−0.657565
$167$	3.52786	0.272994	0.136497	−	0.990640i	$-0.456416\pi$
$167$	3.52786	0.272994	0.136497	+	0.990640i	$0.456416\pi$
$168$	0	0
$169$	0.0901699	0.00693615
$170$	7.76393	0.595466
$171$	0	0
$172$	−4.76393	−0.363246
$173$	14.3262	1.08920	0.544602	−	0.838695i	$-0.316681\pi$
$173$	14.3262	1.08920	0.544602	+	0.838695i	$0.316681\pi$
$174$	0	0
$175$	−3.81966	−0.288739
$176$	−1.23607	−0.0931721
$177$	0	0
$178$	−2.14590	−0.160842
$179$	−3.41641	−0.255354	−0.127677	−	0.991816i	$-0.540752\pi$
$179$	−3.41641	−0.255354	−0.127677	+	0.991816i	$0.540752\pi$
$180$	0	0
$181$	−21.4164	−1.59187	−0.795935	−	0.605383i	$-0.793020\pi$
$181$	−21.4164	−1.59187	−0.795935	+	0.605383i	$0.793020\pi$
$182$	4.47214	0.331497
$183$	0	0
$184$	−0.763932	−0.0563178
$185$	14.6738	1.07884
$186$	0	0
$187$	−6.94427	−0.507815
$188$	4.47214	0.326164
$189$	0	0
$190$	0	0
$191$	−12.4721	−0.902452	−0.451226	−	0.892410i	$-0.649013\pi$
$191$	−12.4721	−0.902452	−0.451226	+	0.892410i	$0.649013\pi$
$192$	0	0
$193$	−20.4721	−1.47362	−0.736808	−	0.676102i	$-0.763668\pi$
$193$	−20.4721	−1.47362	−0.736808	+	0.676102i	$0.763668\pi$
$194$	6.38197	0.458198
$195$	0	0
$196$	−5.47214	−0.390867
$197$	10.8541	0.773323	0.386661	−	0.922222i	$-0.373628\pi$
$197$	10.8541	0.773323	0.386661	+	0.922222i	$0.373628\pi$
$198$	0	0
$199$	1.52786	0.108307	0.0541537	−	0.998533i	$-0.482754\pi$
$199$	1.52786	0.108307	0.0541537	+	0.998533i	$0.482754\pi$
$200$	−3.09017	−0.218508
$201$	0	0
$202$	13.3820	0.941552
$203$	−2.58359	−0.181333
$204$	0	0
$205$	8.09017	0.565042
$206$	−16.6525	−1.16023
$207$	0	0
$208$	3.61803	0.250866
$209$	0	0
$210$	0	0
$211$	−2.76393	−0.190277	−0.0951385	−	0.995464i	$-0.530329\pi$
$211$	−2.76393	−0.190277	−0.0951385	+	0.995464i	$0.530329\pi$
$212$	−5.09017	−0.349594
$213$	0	0
$214$	17.4164	1.19056
$215$	−6.58359	−0.448997
$216$	0	0
$217$	4.00000	0.271538
$218$	18.0344	1.22145
$219$	0	0
$220$	−1.70820	−0.115167
$221$	20.3262	1.36729
$222$	0	0
$223$	14.2918	0.957049	0.478525	−	0.878074i	$-0.341172\pi$
$223$	14.2918	0.957049	0.478525	+	0.878074i	$0.341172\pi$
$224$	1.23607	0.0825883
$225$	0	0
$226$	3.14590	0.209262
$227$	−8.94427	−0.593652	−0.296826	−	0.954932i	$-0.595928\pi$
$227$	−8.94427	−0.593652	−0.296826	+	0.954932i	$0.595928\pi$
$228$	0	0
$229$	6.61803	0.437332	0.218666	−	0.975800i	$-0.429829\pi$
$229$	6.61803	0.437332	0.218666	+	0.975800i	$0.429829\pi$
$230$	−1.05573	−0.0696126
$231$	0	0
$232$	−2.09017	−0.137226
$233$	18.5623	1.21606	0.608029	−	0.793915i	$-0.291961\pi$
$233$	18.5623	1.21606	0.608029	+	0.793915i	$0.291961\pi$
$234$	0	0
$235$	6.18034	0.403161
$236$	−8.47214	−0.551489
$237$	0	0
$238$	6.94427	0.450130
$239$	−7.81966	−0.505812	−0.252906	−	0.967491i	$-0.581386\pi$
$239$	−7.81966	−0.505812	−0.252906	+	0.967491i	$0.581386\pi$
$240$	0	0
$241$	1.41641	0.0912389	0.0456194	−	0.998959i	$-0.485474\pi$
$241$	1.41641	0.0912389	0.0456194	+	0.998959i	$0.485474\pi$
$242$	−9.47214	−0.608892
$243$	0	0
$244$	3.61803	0.231621
$245$	−7.56231	−0.483138
$246$	0	0
$247$	0	0
$248$	3.23607	0.205491
$249$	0	0
$250$	−11.1803	−0.707107
$251$	15.7082	0.991493	0.495747	−	0.868467i	$-0.334895\pi$
$251$	15.7082	0.991493	0.495747	+	0.868467i	$0.334895\pi$
$252$	0	0
$253$	0.944272	0.0593659
$254$	15.4164	0.967311
$255$	0	0
$256$	1.00000	0.0625000
$257$	−20.6180	−1.28612	−0.643059	−	0.765817i	$-0.722335\pi$
$257$	−20.6180	−1.28612	−0.643059	+	0.765817i	$0.722335\pi$
$258$	0	0
$259$	13.1246	0.815524
$260$	5.00000	0.310087
$261$	0	0
$262$	4.47214	0.276289
$263$	14.9443	0.921503	0.460752	−	0.887529i	$-0.347580\pi$
$263$	14.9443	0.921503	0.460752	+	0.887529i	$0.347580\pi$
$264$	0	0
$265$	−7.03444	−0.432122
$266$	0	0
$267$	0	0
$268$	−1.70820	−0.104345
$269$	1.32624	0.0808622	0.0404311	−	0.999182i	$-0.487127\pi$
$269$	1.32624	0.0808622	0.0404311	+	0.999182i	$0.487127\pi$
$270$	0	0
$271$	26.0000	1.57939	0.789694	−	0.613501i	$-0.210239\pi$
$271$	26.0000	1.57939	0.789694	+	0.613501i	$0.210239\pi$
$272$	5.61803	0.340643
$273$	0	0
$274$	−4.61803	−0.278986
$275$	3.81966	0.230334
$276$	0	0
$277$	1.09017	0.0655020	0.0327510	−	0.999464i	$-0.489573\pi$
$277$	1.09017	0.0655020	0.0327510	+	0.999464i	$0.489573\pi$
$278$	−14.0000	−0.839664
$279$	0	0
$280$	1.70820	0.102085
$281$	25.9787	1.54976	0.774880	−	0.632108i	$-0.217810\pi$
$281$	25.9787	1.54976	0.774880	+	0.632108i	$0.217810\pi$
$282$	0	0
$283$	13.5279	0.804148	0.402074	−	0.915607i	$-0.368289\pi$
$283$	13.5279	0.804148	0.402074	+	0.915607i	$0.368289\pi$
$284$	14.9443	0.886779
$285$	0	0
$286$	−4.47214	−0.264443
$287$	7.23607	0.427132
$288$	0	0
$289$	14.5623	0.856606
$290$	−2.88854	−0.169621
$291$	0	0
$292$	−3.38197	−0.197915
$293$	−5.05573	−0.295359	−0.147679	−	0.989035i	$-0.547180\pi$
$293$	−5.05573	−0.295359	−0.147679	+	0.989035i	$0.547180\pi$
$294$	0	0
$295$	−11.7082	−0.681678
$296$	10.6180	0.617161
$297$	0	0
$298$	1.90983	0.110633
$299$	−2.76393	−0.159842
$300$	0	0
$301$	−5.88854	−0.339410
$302$	5.52786	0.318093
$303$	0	0
$304$	0	0
$305$	5.00000	0.286299
$306$	0	0
$307$	−1.70820	−0.0974923	−0.0487462	−	0.998811i	$-0.515523\pi$
$307$	−1.70820	−0.0974923	−0.0487462	+	0.998811i	$0.515523\pi$
$308$	−1.52786	−0.0870581
$309$	0	0
$310$	4.47214	0.254000
$311$	−26.7639	−1.51764	−0.758822	−	0.651298i	$-0.774225\pi$
$311$	−26.7639	−1.51764	−0.758822	+	0.651298i	$0.774225\pi$
$312$	0	0
$313$	22.5066	1.27215	0.636073	−	0.771628i	$-0.280558\pi$
$313$	22.5066	1.27215	0.636073	+	0.771628i	$0.280558\pi$
$314$	11.6180	0.655644
$315$	0	0
$316$	−7.23607	−0.407061
$317$	14.3262	0.804642	0.402321	−	0.915499i	$-0.368204\pi$
$317$	14.3262	0.804642	0.402321	+	0.915499i	$0.368204\pi$
$318$	0	0
$319$	2.58359	0.144653
$320$	1.38197	0.0772542
$321$	0	0
$322$	−0.944272	−0.0526222
$323$	0	0
$324$	0	0
$325$	−11.1803	−0.620174
$326$	−20.6525	−1.14383
$327$	0	0
$328$	5.85410	0.323239
$329$	5.52786	0.304761
$330$	0	0
$331$	12.2918	0.675618	0.337809	−	0.941215i	$-0.390314\pi$
$331$	12.2918	0.675618	0.337809	+	0.941215i	$0.390314\pi$
$332$	−8.47214	−0.464969
$333$	0	0
$334$	3.52786	0.193036
$335$	−2.36068	−0.128978
$336$	0	0
$337$	4.61803	0.251560	0.125780	−	0.992058i	$-0.459857\pi$
$337$	4.61803	0.251560	0.125780	+	0.992058i	$0.459857\pi$
$338$	0.0901699	0.00490460
$339$	0	0
$340$	7.76393	0.421058
$341$	−4.00000	−0.216612
$342$	0	0
$343$	−15.4164	−0.832408
$344$	−4.76393	−0.256854
$345$	0	0
$346$	14.3262	0.770183
$347$	−28.6525	−1.53815	−0.769073	−	0.639161i	$-0.779282\pi$
$347$	−28.6525	−1.53815	−0.769073	+	0.639161i	$0.779282\pi$
$348$	0	0
$349$	−21.9098	−1.17281	−0.586403	−	0.810019i	$-0.699457\pi$
$349$	−21.9098	−1.17281	−0.586403	+	0.810019i	$0.699457\pi$
$350$	−3.81966	−0.204169
$351$	0	0
$352$	−1.23607	−0.0658826
$353$	−18.7426	−0.997570	−0.498785	−	0.866726i	$-0.666220\pi$
$353$	−18.7426	−0.997570	−0.498785	+	0.866726i	$0.666220\pi$
$354$	0	0
$355$	20.6525	1.09612
$356$	−2.14590	−0.113732
$357$	0	0
$358$	−3.41641	−0.180563
$359$	−20.4721	−1.08048	−0.540239	−	0.841512i	$-0.681666\pi$
$359$	−20.4721	−1.08048	−0.540239	+	0.841512i	$0.681666\pi$
$360$	0	0
$361$	0	0
$362$	−21.4164	−1.12562
$363$	0	0
$364$	4.47214	0.234404
$365$	−4.67376	−0.244636
$366$	0	0
$367$	−34.8328	−1.81826	−0.909129	−	0.416514i	$-0.863252\pi$
$367$	−34.8328	−1.81826	−0.909129	+	0.416514i	$0.863252\pi$
$368$	−0.763932	−0.0398227
$369$	0	0
$370$	14.6738	0.762853
$371$	−6.29180	−0.326654
$372$	0	0
$373$	2.67376	0.138442	0.0692211	−	0.997601i	$-0.477949\pi$
$373$	2.67376	0.138442	0.0692211	+	0.997601i	$0.477949\pi$
$374$	−6.94427	−0.359080
$375$	0	0
$376$	4.47214	0.230633
$377$	−7.56231	−0.389479
$378$	0	0
$379$	−8.58359	−0.440910	−0.220455	−	0.975397i	$-0.570754\pi$
$379$	−8.58359	−0.440910	−0.220455	+	0.975397i	$0.570754\pi$
$380$	0	0
$381$	0	0
$382$	−12.4721	−0.638130
$383$	−25.2361	−1.28950	−0.644751	−	0.764392i	$-0.723039\pi$
$383$	−25.2361	−1.28950	−0.644751	+	0.764392i	$0.723039\pi$
$384$	0	0
$385$	−2.11146	−0.107610
$386$	−20.4721	−1.04200
$387$	0	0
$388$	6.38197	0.323995
$389$	11.7984	0.598201	0.299101	−	0.954222i	$-0.403313\pi$
$389$	11.7984	0.598201	0.299101	+	0.954222i	$0.403313\pi$
$390$	0	0
$391$	−4.29180	−0.217045
$392$	−5.47214	−0.276385
$393$	0	0
$394$	10.8541	0.546822
$395$	−10.0000	−0.503155
$396$	0	0
$397$	22.3607	1.12225	0.561125	−	0.827731i	$-0.310369\pi$
$397$	22.3607	1.12225	0.561125	+	0.827731i	$0.310369\pi$
$398$	1.52786	0.0765849
$399$	0	0
$400$	−3.09017	−0.154508
$401$	−1.05573	−0.0527205	−0.0263603	−	0.999653i	$-0.508392\pi$
$401$	−1.05573	−0.0527205	−0.0263603	+	0.999653i	$0.508392\pi$
$402$	0	0
$403$	11.7082	0.583227
$404$	13.3820	0.665778
$405$	0	0
$406$	−2.58359	−0.128222
$407$	−13.1246	−0.650563
$408$	0	0
$409$	−17.2705	−0.853972	−0.426986	−	0.904258i	$-0.640425\pi$
$409$	−17.2705	−0.853972	−0.426986	+	0.904258i	$0.640425\pi$
$410$	8.09017	0.399545
$411$	0	0
$412$	−16.6525	−0.820409
$413$	−10.4721	−0.515300
$414$	0	0
$415$	−11.7082	−0.574733
$416$	3.61803	0.177389
$417$	0	0
$418$	0	0
$419$	−2.47214	−0.120772	−0.0603859	−	0.998175i	$-0.519233\pi$
$419$	−2.47214	−0.120772	−0.0603859	+	0.998175i	$0.519233\pi$
$420$	0	0
$421$	25.0902	1.22282	0.611410	−	0.791314i	$-0.290603\pi$
$421$	25.0902	1.22282	0.611410	+	0.791314i	$0.290603\pi$
$422$	−2.76393	−0.134546
$423$	0	0
$424$	−5.09017	−0.247201
$425$	−17.3607	−0.842117
$426$	0	0
$427$	4.47214	0.216422
$428$	17.4164	0.841854
$429$	0	0
$430$	−6.58359	−0.317489
$431$	24.6525	1.18747	0.593734	−	0.804661i	$-0.297653\pi$
$431$	24.6525	1.18747	0.593734	+	0.804661i	$0.297653\pi$
$432$	0	0
$433$	4.03444	0.193883	0.0969415	−	0.995290i	$-0.469094\pi$
$433$	4.03444	0.193883	0.0969415	+	0.995290i	$0.469094\pi$
$434$	4.00000	0.192006
$435$	0	0
$436$	18.0344	0.863693
$437$	0	0
$438$	0	0
$439$	24.7639	1.18192	0.590959	−	0.806702i	$-0.298749\pi$
$439$	24.7639	1.18192	0.590959	+	0.806702i	$0.298749\pi$
$440$	−1.70820	−0.0814354
$441$	0	0
$442$	20.3262	0.966821
$443$	−4.76393	−0.226341	−0.113171	−	0.993576i	$-0.536101\pi$
$443$	−4.76393	−0.226341	−0.113171	+	0.993576i	$0.536101\pi$
$444$	0	0
$445$	−2.96556	−0.140581
$446$	14.2918	0.676736
$447$	0	0
$448$	1.23607	0.0583987
$449$	−1.79837	−0.0848705	−0.0424353	−	0.999099i	$-0.513512\pi$
$449$	−1.79837	−0.0848705	−0.0424353	+	0.999099i	$0.513512\pi$
$450$	0	0
$451$	−7.23607	−0.340733
$452$	3.14590	0.147971
$453$	0	0
$454$	−8.94427	−0.419775
$455$	6.18034	0.289739
$456$	0	0
$457$	−26.2148	−1.22628	−0.613138	−	0.789976i	$-0.710093\pi$
$457$	−26.2148	−1.22628	−0.613138	+	0.789976i	$0.710093\pi$
$458$	6.61803	0.309240
$459$	0	0
$460$	−1.05573	−0.0492236
$461$	5.41641	0.252267	0.126134	−	0.992013i	$-0.459743\pi$
$461$	5.41641	0.252267	0.126134	+	0.992013i	$0.459743\pi$
$462$	0	0
$463$	−22.9443	−1.06631	−0.533155	−	0.846017i	$-0.678994\pi$
$463$	−22.9443	−1.06631	−0.533155	+	0.846017i	$0.678994\pi$
$464$	−2.09017	−0.0970337
$465$	0	0
$466$	18.5623	0.859882
$467$	−29.3050	−1.35607	−0.678036	−	0.735029i	$-0.737169\pi$
$467$	−29.3050	−1.35607	−0.678036	+	0.735029i	$0.737169\pi$
$468$	0	0
$469$	−2.11146	−0.0974980
$470$	6.18034	0.285078
$471$	0	0
$472$	−8.47214	−0.389962
$473$	5.88854	0.270756
$474$	0	0
$475$	0	0
$476$	6.94427	0.318290
$477$	0	0
$478$	−7.81966	−0.357663
$479$	5.70820	0.260814	0.130407	−	0.991461i	$-0.458372\pi$
$479$	5.70820	0.260814	0.130407	+	0.991461i	$0.458372\pi$
$480$	0	0
$481$	38.4164	1.75164
$482$	1.41641	0.0645156
$483$	0	0
$484$	−9.47214	−0.430552
$485$	8.81966	0.400480
$486$	0	0
$487$	14.7639	0.669018	0.334509	−	0.942393i	$-0.391430\pi$
$487$	14.7639	0.669018	0.334509	+	0.942393i	$0.391430\pi$
$488$	3.61803	0.163781
$489$	0	0
$490$	−7.56231	−0.341630
$491$	10.4721	0.472601	0.236300	−	0.971680i	$-0.424065\pi$
$491$	10.4721	0.472601	0.236300	+	0.971680i	$0.424065\pi$
$492$	0	0
$493$	−11.7426	−0.528862
$494$	0	0
$495$	0	0
$496$	3.23607	0.145304
$497$	18.4721	0.828589
$498$	0	0
$499$	27.3050	1.22234	0.611169	−	0.791500i	$-0.290700\pi$
$499$	27.3050	1.22234	0.611169	+	0.791500i	$0.290700\pi$
$500$	−11.1803	−0.500000
$501$	0	0
$502$	15.7082	0.701091
$503$	−11.2361	−0.500992	−0.250496	−	0.968118i	$-0.580594\pi$
$503$	−11.2361	−0.500992	−0.250496	+	0.968118i	$0.580594\pi$
$504$	0	0
$505$	18.4934	0.822946
$506$	0.944272	0.0419780
$507$	0	0
$508$	15.4164	0.683992
$509$	17.6869	0.783959	0.391979	−	0.919974i	$-0.371790\pi$
$509$	17.6869	0.783959	0.391979	+	0.919974i	$0.371790\pi$
$510$	0	0
$511$	−4.18034	−0.184927
$512$	1.00000	0.0441942
$513$	0	0
$514$	−20.6180	−0.909422
$515$	−23.0132	−1.01408
$516$	0	0
$517$	−5.52786	−0.243115
$518$	13.1246	0.576662
$519$	0	0
$520$	5.00000	0.219265
$521$	−28.2705	−1.23855	−0.619277	−	0.785173i	$-0.712574\pi$
$521$	−28.2705	−1.23855	−0.619277	+	0.785173i	$0.712574\pi$
$522$	0	0
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	4.47214	0.195366
$525$	0	0
$526$	14.9443	0.651601
$527$	18.1803	0.791948
$528$	0	0
$529$	−22.4164	−0.974626
$530$	−7.03444	−0.305557
$531$	0	0
$532$	0	0
$533$	21.1803	0.917422
$534$	0	0
$535$	24.0689	1.04059
$536$	−1.70820	−0.0737832
$537$	0	0
$538$	1.32624	0.0571782
$539$	6.76393	0.291343
$540$	0	0
$541$	−24.3820	−1.04826	−0.524131	−	0.851637i	$-0.675610\pi$
$541$	−24.3820	−1.04826	−0.524131	+	0.851637i	$0.675610\pi$
$542$	26.0000	1.11680
$543$	0	0
$544$	5.61803	0.240871
$545$	24.9230	1.06758
$546$	0	0
$547$	22.9443	0.981026	0.490513	−	0.871434i	$-0.336809\pi$
$547$	22.9443	0.981026	0.490513	+	0.871434i	$0.336809\pi$
$548$	−4.61803	−0.197273
$549$	0	0
$550$	3.81966	0.162871
$551$	0	0
$552$	0	0
$553$	−8.94427	−0.380349
$554$	1.09017	0.0463169
$555$	0	0
$556$	−14.0000	−0.593732
$557$	−22.9443	−0.972180	−0.486090	−	0.873909i	$-0.661577\pi$
$557$	−22.9443	−0.972180	−0.486090	+	0.873909i	$0.661577\pi$
$558$	0	0
$559$	−17.2361	−0.729008
$560$	1.70820	0.0721848
$561$	0	0
$562$	25.9787	1.09585
$563$	−39.7082	−1.67350	−0.836751	−	0.547584i	$-0.815548\pi$
$563$	−39.7082	−1.67350	−0.836751	+	0.547584i	$0.815548\pi$
$564$	0	0
$565$	4.34752	0.182902
$566$	13.5279	0.568619
$567$	0	0
$568$	14.9443	0.627048
$569$	−19.0902	−0.800302	−0.400151	−	0.916449i	$-0.631042\pi$
$569$	−19.0902	−0.800302	−0.400151	+	0.916449i	$0.631042\pi$
$570$	0	0
$571$	34.5410	1.44550	0.722748	−	0.691111i	$-0.242879\pi$
$571$	34.5410	1.44550	0.722748	+	0.691111i	$0.242879\pi$
$572$	−4.47214	−0.186989
$573$	0	0
$574$	7.23607	0.302028
$575$	2.36068	0.0984472
$576$	0	0
$577$	−18.9230	−0.787774	−0.393887	−	0.919159i	$-0.628870\pi$
$577$	−18.9230	−0.787774	−0.393887	+	0.919159i	$0.628870\pi$
$578$	14.5623	0.605712
$579$	0	0
$580$	−2.88854	−0.119940
$581$	−10.4721	−0.434457
$582$	0	0
$583$	6.29180	0.260580
$584$	−3.38197	−0.139947
$585$	0	0
$586$	−5.05573	−0.208850
$587$	−43.1246	−1.77994	−0.889972	−	0.456016i	$-0.849276\pi$
$587$	−43.1246	−1.77994	−0.889972	+	0.456016i	$0.849276\pi$
$588$	0	0
$589$	0	0
$590$	−11.7082	−0.482019
$591$	0	0
$592$	10.6180	0.436399
$593$	−29.9230	−1.22879	−0.614395	−	0.788999i	$-0.710600\pi$
$593$	−29.9230	−1.22879	−0.614395	+	0.788999i	$0.710600\pi$
$594$	0	0
$595$	9.59675	0.393428
$596$	1.90983	0.0782297
$597$	0	0
$598$	−2.76393	−0.113026
$599$	−1.12461	−0.0459504	−0.0229752	−	0.999736i	$-0.507314\pi$
$599$	−1.12461	−0.0459504	−0.0229752	+	0.999736i	$0.507314\pi$
$600$	0	0
$601$	7.52786	0.307068	0.153534	−	0.988143i	$-0.450935\pi$
$601$	7.52786	0.307068	0.153534	+	0.988143i	$0.450935\pi$
$602$	−5.88854	−0.239999
$603$	0	0
$604$	5.52786	0.224926
$605$	−13.0902	−0.532191
$606$	0	0
$607$	23.7771	0.965082	0.482541	−	0.875873i	$-0.339714\pi$
$607$	23.7771	0.965082	0.482541	+	0.875873i	$0.339714\pi$
$608$	0	0
$609$	0	0
$610$	5.00000	0.202444
$611$	16.1803	0.654586
$612$	0	0
$613$	10.6180	0.428858	0.214429	−	0.976740i	$-0.431211\pi$
$613$	10.6180	0.428858	0.214429	+	0.976740i	$0.431211\pi$
$614$	−1.70820	−0.0689375
$615$	0	0
$616$	−1.52786	−0.0615594
$617$	−25.0557	−1.00871	−0.504353	−	0.863498i	$-0.668269\pi$
$617$	−25.0557	−1.00871	−0.504353	+	0.863498i	$0.668269\pi$
$618$	0	0
$619$	−22.3607	−0.898752	−0.449376	−	0.893343i	$-0.648354\pi$
$619$	−22.3607	−0.898752	−0.449376	+	0.893343i	$0.648354\pi$
$620$	4.47214	0.179605
$621$	0	0
$622$	−26.7639	−1.07314
$623$	−2.65248	−0.106269
$624$	0	0
$625$	0	0
$626$	22.5066	0.899544
$627$	0	0
$628$	11.6180	0.463610
$629$	59.6525	2.37850
$630$	0	0
$631$	9.12461	0.363245	0.181623	−	0.983368i	$-0.441865\pi$
$631$	9.12461	0.363245	0.181623	+	0.983368i	$0.441865\pi$
$632$	−7.23607	−0.287835
$633$	0	0
$634$	14.3262	0.568968
$635$	21.3050	0.845461
$636$	0	0
$637$	−19.7984	−0.784440
$638$	2.58359	0.102285
$639$	0	0
$640$	1.38197	0.0546270
$641$	−20.2705	−0.800637	−0.400319	−	0.916376i	$-0.631101\pi$
$641$	−20.2705	−0.800637	−0.400319	+	0.916376i	$0.631101\pi$
$642$	0	0
$643$	14.8328	0.584949	0.292475	−	0.956273i	$-0.405521\pi$
$643$	14.8328	0.584949	0.292475	+	0.956273i	$0.405521\pi$
$644$	−0.944272	−0.0372095
$645$	0	0
$646$	0	0
$647$	32.0689	1.26076	0.630379	−	0.776288i	$-0.282900\pi$
$647$	32.0689	1.26076	0.630379	+	0.776288i	$0.282900\pi$
$648$	0	0
$649$	10.4721	0.411067
$650$	−11.1803	−0.438529
$651$	0	0
$652$	−20.6525	−0.808813
$653$	−38.3951	−1.50252	−0.751259	−	0.660008i	$-0.770553\pi$
$653$	−38.3951	−1.50252	−0.751259	+	0.660008i	$0.770553\pi$
$654$	0	0
$655$	6.18034	0.241486
$656$	5.85410	0.228564
$657$	0	0
$658$	5.52786	0.215499
$659$	−18.5410	−0.722256	−0.361128	−	0.932516i	$-0.617608\pi$
$659$	−18.5410	−0.722256	−0.361128	+	0.932516i	$0.617608\pi$
$660$	0	0
$661$	5.41641	0.210674	0.105337	−	0.994437i	$-0.466408\pi$
$661$	5.41641	0.210674	0.105337	+	0.994437i	$0.466408\pi$
$662$	12.2918	0.477734
$663$	0	0
$664$	−8.47214	−0.328783
$665$	0	0
$666$	0	0
$667$	1.59675	0.0618263
$668$	3.52786	0.136497
$669$	0	0
$670$	−2.36068	−0.0912010
$671$	−4.47214	−0.172645
$672$	0	0
$673$	32.6180	1.25733	0.628666	−	0.777675i	$-0.283601\pi$
$673$	32.6180	1.25733	0.628666	+	0.777675i	$0.283601\pi$
$674$	4.61803	0.177880
$675$	0	0
$676$	0.0901699	0.00346807
$677$	4.38197	0.168413	0.0842063	−	0.996448i	$-0.473165\pi$
$677$	4.38197	0.168413	0.0842063	+	0.996448i	$0.473165\pi$
$678$	0	0
$679$	7.88854	0.302735
$680$	7.76393	0.297733
$681$	0	0
$682$	−4.00000	−0.153168
$683$	−2.18034	−0.0834284	−0.0417142	−	0.999130i	$-0.513282\pi$
$683$	−2.18034	−0.0834284	−0.0417142	+	0.999130i	$0.513282\pi$
$684$	0	0
$685$	−6.38197	−0.243842
$686$	−15.4164	−0.588601
$687$	0	0
$688$	−4.76393	−0.181623
$689$	−18.4164	−0.701609
$690$	0	0
$691$	22.3607	0.850640	0.425320	−	0.905043i	$-0.360161\pi$
$691$	22.3607	0.850640	0.425320	+	0.905043i	$0.360161\pi$
$692$	14.3262	0.544602
$693$	0	0
$694$	−28.6525	−1.08763
$695$	−19.3475	−0.733893
$696$	0	0
$697$	32.8885	1.24574
$698$	−21.9098	−0.829299
$699$	0	0
$700$	−3.81966	−0.144370
$701$	39.6312	1.49685	0.748425	−	0.663220i	$-0.230811\pi$
$701$	39.6312	1.49685	0.748425	+	0.663220i	$0.230811\pi$
$702$	0	0
$703$	0	0
$704$	−1.23607	−0.0465861
$705$	0	0
$706$	−18.7426	−0.705389
$707$	16.5410	0.622089
$708$	0	0
$709$	18.0902	0.679391	0.339695	−	0.940536i	$-0.389676\pi$
$709$	18.0902	0.679391	0.339695	+	0.940536i	$0.389676\pi$
$710$	20.6525	0.775074
$711$	0	0
$712$	−2.14590	−0.0804209
$713$	−2.47214	−0.0925822
$714$	0	0
$715$	−6.18034	−0.231132
$716$	−3.41641	−0.127677
$717$	0	0
$718$	−20.4721	−0.764013
$719$	−25.7082	−0.958754	−0.479377	−	0.877609i	$-0.659137\pi$
$719$	−25.7082	−0.958754	−0.479377	+	0.877609i	$0.659137\pi$
$720$	0	0
$721$	−20.5836	−0.766573
$722$	0	0
$723$	0	0
$724$	−21.4164	−0.795935
$725$	6.45898	0.239881
$726$	0	0
$727$	−13.0557	−0.484210	−0.242105	−	0.970250i	$-0.577838\pi$
$727$	−13.0557	−0.484210	−0.242105	+	0.970250i	$0.577838\pi$
$728$	4.47214	0.165748
$729$	0	0
$730$	−4.67376	−0.172984
$731$	−26.7639	−0.989900
$732$	0	0
$733$	10.0902	0.372689	0.186344	−	0.982484i	$-0.440336\pi$
$733$	10.0902	0.372689	0.186344	+	0.982484i	$0.440336\pi$
$734$	−34.8328	−1.28570
$735$	0	0
$736$	−0.763932	−0.0281589
$737$	2.11146	0.0777765
$738$	0	0
$739$	−31.1246	−1.14494	−0.572469	−	0.819927i	$-0.694014\pi$
$739$	−31.1246	−1.14494	−0.572469	+	0.819927i	$0.694014\pi$
$740$	14.6738	0.539418
$741$	0	0
$742$	−6.29180	−0.230979
$743$	−41.8885	−1.53674	−0.768371	−	0.640005i	$-0.778932\pi$
$743$	−41.8885	−1.53674	−0.768371	+	0.640005i	$0.778932\pi$
$744$	0	0
$745$	2.63932	0.0966972
$746$	2.67376	0.0978934
$747$	0	0
$748$	−6.94427	−0.253908
$749$	21.5279	0.786611
$750$	0	0
$751$	34.0689	1.24319	0.621596	−	0.783338i	$-0.286485\pi$
$751$	34.0689	1.24319	0.621596	+	0.783338i	$0.286485\pi$
$752$	4.47214	0.163082
$753$	0	0
$754$	−7.56231	−0.275403
$755$	7.63932	0.278023
$756$	0	0
$757$	24.5623	0.892732	0.446366	−	0.894850i	$-0.352718\pi$
$757$	24.5623	0.892732	0.446366	+	0.894850i	$0.352718\pi$
$758$	−8.58359	−0.311770
$759$	0	0
$760$	0	0
$761$	18.3607	0.665574	0.332787	−	0.943002i	$-0.392011\pi$
$761$	18.3607	0.665574	0.332787	+	0.943002i	$0.392011\pi$
$762$	0	0
$763$	22.2918	0.807017
$764$	−12.4721	−0.451226
$765$	0	0
$766$	−25.2361	−0.911816
$767$	−30.6525	−1.10680
$768$	0	0
$769$	−3.74265	−0.134963	−0.0674816	−	0.997721i	$-0.521496\pi$
$769$	−3.74265	−0.134963	−0.0674816	+	0.997721i	$0.521496\pi$
$770$	−2.11146	−0.0760916
$771$	0	0
$772$	−20.4721	−0.736808
$773$	14.8541	0.534265	0.267132	−	0.963660i	$-0.413924\pi$
$773$	14.8541	0.534265	0.267132	+	0.963660i	$0.413924\pi$
$774$	0	0
$775$	−10.0000	−0.359211
$776$	6.38197	0.229099
$777$	0	0
$778$	11.7984	0.422992
$779$	0	0
$780$	0	0
$781$	−18.4721	−0.660985
$782$	−4.29180	−0.153474
$783$	0	0
$784$	−5.47214	−0.195433
$785$	16.0557	0.573054
$786$	0	0
$787$	19.5279	0.696093	0.348047	−	0.937477i	$-0.386845\pi$
$787$	19.5279	0.696093	0.348047	+	0.937477i	$0.386845\pi$
$788$	10.8541	0.386661
$789$	0	0
$790$	−10.0000	−0.355784
$791$	3.88854	0.138261
$792$	0	0
$793$	13.0902	0.464846
$794$	22.3607	0.793551
$795$	0	0
$796$	1.52786	0.0541537
$797$	−48.5623	−1.72017	−0.860083	−	0.510155i	$-0.829588\pi$
$797$	−48.5623	−1.72017	−0.860083	+	0.510155i	$0.829588\pi$
$798$	0	0
$799$	25.1246	0.888845
$800$	−3.09017	−0.109254
$801$	0	0
$802$	−1.05573	−0.0372791
$803$	4.18034	0.147521
$804$	0	0
$805$	−1.30495	−0.0459935
$806$	11.7082	0.412404
$807$	0	0
$808$	13.3820	0.470776
$809$	39.7426	1.39728	0.698639	−	0.715475i	$-0.253790\pi$
$809$	39.7426	1.39728	0.698639	+	0.715475i	$0.253790\pi$
$810$	0	0
$811$	−43.5967	−1.53089	−0.765444	−	0.643502i	$-0.777481\pi$
$811$	−43.5967	−1.53089	−0.765444	+	0.643502i	$0.777481\pi$
$812$	−2.58359	−0.0906663
$813$	0	0
$814$	−13.1246	−0.460017
$815$	−28.5410	−0.999748
$816$	0	0
$817$	0	0
$818$	−17.2705	−0.603849
$819$	0	0
$820$	8.09017	0.282521
$821$	14.2705	0.498044	0.249022	−	0.968498i	$-0.419891\pi$
$821$	14.2705	0.498044	0.249022	+	0.968498i	$0.419891\pi$
$822$	0	0
$823$	32.8328	1.14448	0.572240	−	0.820086i	$-0.306075\pi$
$823$	32.8328	1.14448	0.572240	+	0.820086i	$0.306075\pi$
$824$	−16.6525	−0.580116
$825$	0	0
$826$	−10.4721	−0.364372
$827$	−36.5410	−1.27066	−0.635328	−	0.772243i	$-0.719135\pi$
$827$	−36.5410	−1.27066	−0.635328	+	0.772243i	$0.719135\pi$
$828$	0	0
$829$	−43.2705	−1.50285	−0.751423	−	0.659820i	$-0.770632\pi$
$829$	−43.2705	−1.50285	−0.751423	+	0.659820i	$0.770632\pi$
$830$	−11.7082	−0.406398
$831$	0	0
$832$	3.61803	0.125433
$833$	−30.7426	−1.06517
$834$	0	0
$835$	4.87539	0.168720
$836$	0	0
$837$	0	0
$838$	−2.47214	−0.0853985
$839$	−22.0000	−0.759524	−0.379762	−	0.925084i	$-0.623994\pi$
$839$	−22.0000	−0.759524	−0.379762	+	0.925084i	$0.623994\pi$
$840$	0	0
$841$	−24.6312	−0.849351
$842$	25.0902	0.864664
$843$	0	0
$844$	−2.76393	−0.0951385
$845$	0.124612	0.00428678
$846$	0	0
$847$	−11.7082	−0.402299
$848$	−5.09017	−0.174797
$849$	0	0
$850$	−17.3607	−0.595466
$851$	−8.11146	−0.278057
$852$	0	0
$853$	−17.9098	−0.613221	−0.306610	−	0.951835i	$-0.599195\pi$
$853$	−17.9098	−0.613221	−0.306610	+	0.951835i	$0.599195\pi$
$854$	4.47214	0.153033
$855$	0	0
$856$	17.4164	0.595281
$857$	14.5623	0.497439	0.248719	−	0.968576i	$-0.419990\pi$
$857$	14.5623	0.497439	0.248719	+	0.968576i	$0.419990\pi$
$858$	0	0
$859$	−30.0689	−1.02594	−0.512969	−	0.858407i	$-0.671454\pi$
$859$	−30.0689	−1.02594	−0.512969	+	0.858407i	$0.671454\pi$
$860$	−6.58359	−0.224499
$861$	0	0
$862$	24.6525	0.839667
$863$	−9.88854	−0.336610	−0.168305	−	0.985735i	$-0.553829\pi$
$863$	−9.88854	−0.336610	−0.168305	+	0.985735i	$0.553829\pi$
$864$	0	0
$865$	19.7984	0.673165
$866$	4.03444	0.137096
$867$	0	0
$868$	4.00000	0.135769
$869$	8.94427	0.303414
$870$	0	0
$871$	−6.18034	−0.209413
$872$	18.0344	0.610723
$873$	0	0
$874$	0	0
$875$	−13.8197	−0.467190
$876$	0	0
$877$	−40.7426	−1.37578	−0.687891	−	0.725814i	$-0.741463\pi$
$877$	−40.7426	−1.37578	−0.687891	+	0.725814i	$0.741463\pi$
$878$	24.7639	0.835742
$879$	0	0
$880$	−1.70820	−0.0575835
$881$	46.2148	1.55702	0.778508	−	0.627635i	$-0.215977\pi$
$881$	46.2148	1.55702	0.778508	+	0.627635i	$0.215977\pi$
$882$	0	0
$883$	−40.1803	−1.35218	−0.676088	−	0.736821i	$-0.736326\pi$
$883$	−40.1803	−1.35218	−0.676088	+	0.736821i	$0.736326\pi$
$884$	20.3262	0.683645
$885$	0	0
$886$	−4.76393	−0.160047
$887$	10.9443	0.367473	0.183736	−	0.982976i	$-0.441181\pi$
$887$	10.9443	0.367473	0.183736	+	0.982976i	$0.441181\pi$
$888$	0	0
$889$	19.0557	0.639109
$890$	−2.96556	−0.0994057
$891$	0	0
$892$	14.2918	0.478525
$893$	0	0
$894$	0	0
$895$	−4.72136	−0.157818
$896$	1.23607	0.0412941
$897$	0	0
$898$	−1.79837	−0.0600125
$899$	−6.76393	−0.225590
$900$	0	0
$901$	−28.5967	−0.952696
$902$	−7.23607	−0.240935
$903$	0	0
$904$	3.14590	0.104631
$905$	−29.5967	−0.983829
$906$	0	0
$907$	−13.7082	−0.455173	−0.227587	−	0.973758i	$-0.573084\pi$
$907$	−13.7082	−0.455173	−0.227587	+	0.973758i	$0.573084\pi$
$908$	−8.94427	−0.296826
$909$	0	0
$910$	6.18034	0.204876
$911$	46.0689	1.52633	0.763165	−	0.646204i	$-0.223644\pi$
$911$	46.0689	1.52633	0.763165	+	0.646204i	$0.223644\pi$
$912$	0	0
$913$	10.4721	0.346577
$914$	−26.2148	−0.867108
$915$	0	0
$916$	6.61803	0.218666
$917$	5.52786	0.182546
$918$	0	0
$919$	10.1115	0.333546	0.166773	−	0.985995i	$-0.446665\pi$
$919$	10.1115	0.333546	0.166773	+	0.985995i	$0.446665\pi$
$920$	−1.05573	−0.0348063
$921$	0	0
$922$	5.41641	0.178380
$923$	54.0689	1.77970
$924$	0	0
$925$	−32.8115	−1.07884
$926$	−22.9443	−0.753996
$927$	0	0
$928$	−2.09017	−0.0686132
$929$	−10.1591	−0.333308	−0.166654	−	0.986015i	$-0.553296\pi$
$929$	−10.1591	−0.333308	−0.166654	+	0.986015i	$0.553296\pi$
$930$	0	0
$931$	0	0
$932$	18.5623	0.608029
$933$	0	0
$934$	−29.3050	−0.958887
$935$	−9.59675	−0.313847
$936$	0	0
$937$	−19.8885	−0.649730	−0.324865	−	0.945760i	$-0.605319\pi$
$937$	−19.8885	−0.649730	−0.324865	+	0.945760i	$0.605319\pi$
$938$	−2.11146	−0.0689415
$939$	0	0
$940$	6.18034	0.201580
$941$	−26.0000	−0.847576	−0.423788	−	0.905761i	$-0.639300\pi$
$941$	−26.0000	−0.847576	−0.423788	+	0.905761i	$0.639300\pi$
$942$	0	0
$943$	−4.47214	−0.145633
$944$	−8.47214	−0.275745
$945$	0	0
$946$	5.88854	0.191453
$947$	14.1803	0.460799	0.230400	−	0.973096i	$-0.425997\pi$
$947$	14.1803	0.460799	0.230400	+	0.973096i	$0.425997\pi$
$948$	0	0
$949$	−12.2361	−0.397200
$950$	0	0
$951$	0	0
$952$	6.94427	0.225065
$953$	52.8115	1.71073	0.855367	−	0.518023i	$-0.173332\pi$
$953$	52.8115	1.71073	0.855367	+	0.518023i	$0.173332\pi$
$954$	0	0
$955$	−17.2361	−0.557746
$956$	−7.81966	−0.252906
$957$	0	0
$958$	5.70820	0.184424
$959$	−5.70820	−0.184328
$960$	0	0
$961$	−20.5279	−0.662189
$962$	38.4164	1.23859
$963$	0	0
$964$	1.41641	0.0456194
$965$	−28.2918	−0.910745
$966$	0	0
$967$	−44.8328	−1.44173	−0.720863	−	0.693077i	$-0.756254\pi$
$967$	−44.8328	−1.44173	−0.720863	+	0.693077i	$0.756254\pi$
$968$	−9.47214	−0.304446
$969$	0	0
$970$	8.81966	0.283182
$971$	6.54102	0.209911	0.104956	−	0.994477i	$-0.466530\pi$
$971$	6.54102	0.209911	0.104956	+	0.994477i	$0.466530\pi$
$972$	0	0
$973$	−17.3050	−0.554771
$974$	14.7639	0.473067
$975$	0	0
$976$	3.61803	0.115810
$977$	−20.0344	−0.640959	−0.320479	−	0.947256i	$-0.603844\pi$
$977$	−20.0344	−0.640959	−0.320479	+	0.947256i	$0.603844\pi$
$978$	0	0
$979$	2.65248	0.0847735
$980$	−7.56231	−0.241569
$981$	0	0
$982$	10.4721	0.334179
$983$	29.5967	0.943990	0.471995	−	0.881601i	$-0.343534\pi$
$983$	29.5967	0.943990	0.471995	+	0.881601i	$0.343534\pi$
$984$	0	0
$985$	15.0000	0.477940
$986$	−11.7426	−0.373962
$987$	0	0
$988$	0	0
$989$	3.63932	0.115724
$990$	0	0
$991$	0.652476	0.0207266	0.0103633	−	0.999946i	$-0.496701\pi$
$991$	0.652476	0.0207266	0.0103633	+	0.999946i	$0.496701\pi$
$992$	3.23607	0.102745
$993$	0	0
$994$	18.4721	0.585901
$995$	2.11146	0.0669377
$996$	0	0
$997$	24.5066	0.776131	0.388066	−	0.921632i	$-0.373143\pi$
$997$	24.5066	0.776131	0.388066	+	0.921632i	$0.373143\pi$
$998$	27.3050	0.864323
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6498.2.a.bk.1.1		2
3.2	odd	2		722.2.a.h.1.2	✓	2
12.11	even	2		5776.2.a.t.1.1		2
19.18	odd	2		6498.2.a.be.1.1		2
57.2	even	18		722.2.e.p.99.1		12
57.5	odd	18		722.2.e.q.595.1		12
57.8	even	6		722.2.c.h.653.2		4
57.11	odd	6		722.2.c.i.653.1		4
57.14	even	18		722.2.e.p.595.2		12
57.17	odd	18		722.2.e.q.99.2		12
57.23	odd	18		722.2.e.q.415.1		12
57.26	odd	6		722.2.c.i.429.1		4
57.29	even	18		722.2.e.p.423.1		12
57.32	even	18		722.2.e.p.245.1		12
57.35	odd	18		722.2.e.q.389.2		12
57.41	even	18		722.2.e.p.389.1		12
57.44	odd	18		722.2.e.q.245.2		12
57.47	odd	18		722.2.e.q.423.2		12
57.50	even	6		722.2.c.h.429.2		4
57.53	even	18		722.2.e.p.415.2		12
57.56	even	2		722.2.a.i.1.1	yes	2
228.227	odd	2		5776.2.a.be.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
722.2.a.h.1.2	✓	2	3.2	odd	2
722.2.a.i.1.1	yes	2	57.56	even	2
722.2.c.h.429.2		4	57.50	even	6
722.2.c.h.653.2		4	57.8	even	6
722.2.c.i.429.1		4	57.26	odd	6
722.2.c.i.653.1		4	57.11	odd	6
722.2.e.p.99.1		12	57.2	even	18
722.2.e.p.245.1		12	57.32	even	18
722.2.e.p.389.1		12	57.41	even	18
722.2.e.p.415.2		12	57.53	even	18
722.2.e.p.423.1		12	57.29	even	18
722.2.e.p.595.2		12	57.14	even	18
722.2.e.q.99.2		12	57.17	odd	18
722.2.e.q.245.2		12	57.44	odd	18
722.2.e.q.389.2		12	57.35	odd	18
722.2.e.q.415.1		12	57.23	odd	18
722.2.e.q.423.2		12	57.47	odd	18
722.2.e.q.595.1		12	57.5	odd	18
5776.2.a.t.1.1		2	12.11	even	2
5776.2.a.be.1.2		2	228.227	odd	2
6498.2.a.be.1.1		2	19.18	odd	2
6498.2.a.bk.1.1		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6498 = 2 \cdot 3^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6498.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.38197 q^{5} +1.23607 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.38197 q^{5} +1.23607 q^{7} +1.00000 q^{8} +1.38197 q^{10} -1.23607 q^{11} +3.61803 q^{13} +1.23607 q^{14} +1.00000 q^{16} +5.61803 q^{17} +1.38197 q^{20} -1.23607 q^{22} -0.763932 q^{23} -3.09017 q^{25} +3.61803 q^{26} +1.23607 q^{28} -2.09017 q^{29} +3.23607 q^{31} +1.00000 q^{32} +5.61803 q^{34} +1.70820 q^{35} +10.6180 q^{37} +1.38197 q^{40} +5.85410 q^{41} -4.76393 q^{43} -1.23607 q^{44} -0.763932 q^{46} +4.47214 q^{47} -5.47214 q^{49} -3.09017 q^{50} +3.61803 q^{52} -5.09017 q^{53} -1.70820 q^{55} +1.23607 q^{56} -2.09017 q^{58} -8.47214 q^{59} +3.61803 q^{61} +3.23607 q^{62} +1.00000 q^{64} +5.00000 q^{65} -1.70820 q^{67} +5.61803 q^{68} +1.70820 q^{70} +14.9443 q^{71} -3.38197 q^{73} +10.6180 q^{74} -1.52786 q^{77} -7.23607 q^{79} +1.38197 q^{80} +5.85410 q^{82} -8.47214 q^{83} +7.76393 q^{85} -4.76393 q^{86} -1.23607 q^{88} -2.14590 q^{89} +4.47214 q^{91} -0.763932 q^{92} +4.47214 q^{94} +6.38197 q^{97} -5.47214 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 5 q^{5} - 2 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + 5 * q^5 - 2 * q^7 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} + 5 q^{5} - 2 q^{7} + 2 q^{8} + 5 q^{10} + 2 q^{11} + 5 q^{13} - 2 q^{14} + 2 q^{16} + 9 q^{17} + 5 q^{20} + 2 q^{22} - 6 q^{23} + 5 q^{25} + 5 q^{26} - 2 q^{28} + 7 q^{29} + 2 q^{31} + 2 q^{32} + 9 q^{34} - 10 q^{35} + 19 q^{37} + 5 q^{40} + 5 q^{41} - 14 q^{43} + 2 q^{44} - 6 q^{46} - 2 q^{49} + 5 q^{50} + 5 q^{52} + q^{53} + 10 q^{55} - 2 q^{56} + 7 q^{58} - 8 q^{59} + 5 q^{61} + 2 q^{62} + 2 q^{64} + 10 q^{65} + 10 q^{67} + 9 q^{68} - 10 q^{70} + 12 q^{71} - 9 q^{73} + 19 q^{74} - 12 q^{77} - 10 q^{79} + 5 q^{80} + 5 q^{82} - 8 q^{83} + 20 q^{85} - 14 q^{86} + 2 q^{88} - 11 q^{89} - 6 q^{92} + 15 q^{97} - 2 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 5 * q^5 - 2 * q^7 + 2 * q^8 + 5 * q^10 + 2 * q^11 + 5 * q^13 - 2 * q^14 + 2 * q^16 + 9 * q^17 + 5 * q^20 + 2 * q^22 - 6 * q^23 + 5 * q^25 + 5 * q^26 - 2 * q^28 + 7 * q^29 + 2 * q^31 + 2 * q^32 + 9 * q^34 - 10 * q^35 + 19 * q^37 + 5 * q^40 + 5 * q^41 - 14 * q^43 + 2 * q^44 - 6 * q^46 - 2 * q^49 + 5 * q^50 + 5 * q^52 + q^53 + 10 * q^55 - 2 * q^56 + 7 * q^58 - 8 * q^59 + 5 * q^61 + 2 * q^62 + 2 * q^64 + 10 * q^65 + 10 * q^67 + 9 * q^68 - 10 * q^70 + 12 * q^71 - 9 * q^73 + 19 * q^74 - 12 * q^77 - 10 * q^79 + 5 * q^80 + 5 * q^82 - 8 * q^83 + 20 * q^85 - 14 * q^86 + 2 * q^88 - 11 * q^89 - 6 * q^92 + 15 * q^97 - 2 * q^98

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